AUTHORS: Ivanka Milosevic, Milan Damnjanovic
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ABSTRACT: We show that modified Wigner projector technique and generalized Bloch theorem approach yield maximally efficient diagonalization of the Hamiltonian of the large symmetrical systems. For the sake of illustration, we perform a case study of the simplified DNA molecule model and solve the energy eigenproblem analytically by using the unit symmetry cell (symcell) and the corresponding low-dimensional subspaces only. Relevant dynamical parameters are automatically obtained, enabling direct interpretation of the result. Effectiveness of the procedure is based on the two key points: (1) replacing infinite sums over the group elements by modified group projectors which are inherently determined by the group generators only; (2) reducing the dynamics of the system (from the infinite dimensional state space) to the low-dimensional symcell subspace, taking the benefit from the induced structure of the state space. Unlike the original Wigner projectors, the modified group projector technique is directly numerically applicable.
KEYWORDS: Deoxyribonucleic Acid (DNA), Symmetry,Wigner Group Projectors, Modified Group Projector Technique, Generalized Bloch Theorem, Electronic Bands, Eigenproblem, Inductive Spaces, Tight-binding Approximation
REFERENCES:
[1] E. P. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York 1959
[2] M. Damnjanovic and I. Milo ´ sevi ˇ c, Full sym- ´ metry implementation in condensed matter and molecular physics – Modified group projector technique, Phys. Rep. 581, 2015, pp. 1-43.
[3] M. Damnjanovic and I. Milo ´ sevi ˇ c,´ Line Groups in Physics - Theory and Applications to Nanotubes and Polymers, Springer, Berlin Heidelberg 2010
[4] N. W. Ashcroft AND N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York 1976
[5] J. D. Watson and F. H. C. Crick, A Structure for Deoxyribose Nucleic Acid, Nature 171, 1953, pp. 737-738.
[6] X. Zhao, J. Cui, Z. Li, X. Xu, Z. Shang, Y. Li, G. Wang, and R. Li, Symmetries of deoxyribonucleic acid (DNA) and related molecules J. Math. Chem. 2016, pp.1-33.
[7] J. C. Wang, Helical repeat of DNA in solution, PNAS 76, 1979, pp. 200-203.
[8] I. Milosevi ˇ c, A.Damjanovi ´ c and M. Damn- ´ janovic, Ch. XIV. In ´ Quantum Mechanical Simulation Methods in Studying Biological Systems, ed. D. Bicout and M. Field, Springer-Verlag, Berlin Heidelberg New York 1996
